Why is 2y * dy/dx = 1 when y = sqrt of x

Question

I saw this as one of the answers to $y=\sqrt{x}$ but don't get the $2y$ part

\begin{align} y&=\sqrt{x}\\ y^2&=x\\ 2y\frac{\mathrm{d}y}{\mathrm{d}x}&=1\\ \frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{1}{2y}\\ &=\frac{1}{2\sqrt{x}}\tag{2} \end{align}

Answer 1

It follows by the chain rule: $\frac{d}{dx}y^2=\frac{dy^2}{dy}\frac{dy}{dx}=2y\frac{dy}{dx}$.

Answer 2

We have that

$$y=\sqrt x \implies \frac{dy}{dx} = \frac1{2\sqrt x}$$

therefore

$$2y\frac{dy}{dx}=2\sqrt x\cdot \frac1{2\sqrt x}=1$$

Answer 3

So $y=y(x)$ thus we cannot differentiate it explicitly; we can differentiate it implicitly. \begin{align} \frac{d}{dx}y^2 &= \frac{d}{dy}\frac{dy}{dx}y^2 \\ &=\frac{d}{dy}y^2\frac{dy}{dx} \\ &=2y \frac{dy}{dx} \end{align}

Answer 4

$$y^2 =x$$

Differentiating both sides with respect to $x$,

$$\frac{d(y^2)}{dx} = \frac{dx}{dx}$$

Using chain rule,

$$\begin{align}\frac{d(y^2)}{dx} & = \frac{d(y^2)}{dy} \times\frac{d(y)}{dx} \\ \frac{d(y^2)}{dx} &= 2y \space \frac{dy}{dx} \end{align}$$

Substituting this into the original equation ,

$$\begin{align}2y \space \frac{dy}{dx} = 1 \implies \frac{dy}{dx} & =\frac {1}{2y} \implies \boxed{\frac {dy}{dx} = \color{blue}{\frac{1}{2\sqrt x}}}\end{align}$$